Question: Simplify the following expression: $r = \dfrac{84z^2 - 24z}{12z^2 + 48z}$ You can assume $z \neq 0$.
Solution: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $84z^2 - 24z = (2\cdot2\cdot3\cdot7 \cdot z \cdot z) - (2\cdot2\cdot2\cdot3 \cdot z)$ The denominator can be factored: $12z^2 + 48z = (2\cdot2\cdot3 \cdot z \cdot z) + (2\cdot2\cdot2\cdot2\cdot3 \cdot z)$ The greatest common factor of all the terms is $12z$ Factoring out $12z$ gives us: $r = \dfrac{(12z)(7z - 2)}{(12z)(z + 4)}$ Dividing both the numerator and denominator by $12z$ gives: $r = \dfrac{7z - 2}{z + 4}$